
where \(\gamma _1>0\)  and \(\gamma _2>0\)  denote the gain in magnetic materials [1]}, [2]} and \(a_{\textmd {\textmd {in}}}\) , \(b_{\textmd {\textmd {in}}}\)  and \(c_{\textmd {\textmd {in}}}\)  are the noise operators, which satisfy the following properties
\(\langle a_{\textmd {\textmd {in}}}^\dagger (t)a_{\textmd {\textmd {in}}}(t^{\prime })\rangle =0, \langle a_{\textmd {\textmd {in}}}(t)a_{\textmd {\textmd {in}}}^\dagger (t^{\prime })\rangle =\delta (t-t^{\prime }),\\\langle b_{\textmd {\textmd {in}}}^\dagger (t)b_{\textmd {\textmd {in}}}(t^{\prime })\rangle =0, \langle b_{\textmd {\textmd {in}}}(t)b_{\textmd {\textmd {in}}}^\dagger (t^{\prime })\rangle =\delta (t-t^{\prime }),\\\langle c_{\textmd {\textmd {in}}}^\dagger (t)c_{\textmd {\textmd {in}}}(t^{\prime })\rangle =0, \langle c_{\textmd {\textmd {in}}}(t)c_{\textmd {\textmd {in}}}^\dagger (t^{\prime })\rangle =\delta (t-t^{\prime }).\) 
